Question

Part A - Describe the transformation of f(x) to h(x) if h(x) = f(x + 5) – 2.

Part B - If f(x) = 3x² – 2, find h(x).
Part C - Evaluate h(-3).

Answer 1

The transformation of f(x) to h(x) consists of shifting the graph to the left by 5 units and downward by 2 units. The function h(x) is found to be 3(x + 5)² - 4, and evaluating h(-3) yields a value of 8.

Step-by-step explanation:

Part A: Transformation Description

The transformation from f(x) to h(x) = f(x + 5) - 2 can be described as a horizontal shift to the left by 5 units and a vertical shift downward by 2 units. This is due to the fact that adding a positive number inside the function argument (x + 5) shifts the graph to the left, while subtracting a constant from the function (-2) shifts the graph downward.

Part B: Finding h(x)

To find h(x) when f(x) = 3x² - 2, we substitute x + 5 for x in f(x) and then subtract 2:
h(x) = 3(x + 5)² - 2 - 2 = 3(x + 5)² - 4. This represents the new function h(x) after the transformation.

Part C: Evaluate h(-3)

For the calculation of h(-3), we use the found expression of h(x) and substitute x with -3:
h(-3) = 3(-3 + 5)² - 4 = 3(2)² - 4 = 3(4) - 4 = 12 - 4 = 8. Hence, h(-3) evaluates to 8.